Computing phylogenetic diversity for split systems

In conservation biology, it is a central problem to measure, predict, and preserve biodiversity as species face extinction. In 1992, Faith proposed measuring the diversity of a collection of species in terms of their relationships on a phylogenetic tree and using this information to identify collections of species with high diversity. Here, we are interested in some variants of the resulting optimization problem that arise when considering species whose evolution is better represented by a network rather than a tree. More specifically, we consider the problem of computing phylogenetic diversity relative to a split system on a collection of species of size n. We show that, for general split systems, this problem is NP-hard. In addition, we provide some efficient algorithms for some special classes of split systems, in particular presenting an optimal O(n) time algorithm for phylogenetic trees and an O(n log n + nk) time algorithm for choosing an optimal subset of size k relative to a circular split system.